Carl Graham

Stochastic simulation and Monte-Carlo methods

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Poisson Processes as Particular Markov Processes

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Discretization of Stochastic Differential Equations

We first introduce some practical and theoretical issues of modeling by means of Markov processes. Point processes are introduced in order to model jump instants. The Poisson process is then characterized as a point process without memory. The rest of the chapter consists in its rather detailed study, including various results concerning its simulation and approximation. This study is essential to understand the abstract constructions and the simulation methods for jump Markov processes developed in the following chapters.

Variance Reduction and Stochastic Differential Equations

This chapter develops discretization schemes for stochastic differential equations and their applications to the probabilistic numerical resolution of deterministic parabolic partial differential equations. It starts with some important properties of Ito’s Brownian stochastic calculus, and the existence and uniqueness theorem for stochastic differential equations with Lipschitz coefficients. Then, using probabilistic techniques only, existence, uniqueness, and smoothness properties are proved for solutions of parabolic partial differential equations. To this end, we show that stochastic differential equations with smooth coefficients define stochastic flows, and we prove some properties of such flows. We are then in a position to prove an optimal convergence rate result for the discretization schemes.

Strong Law of Large Numbers and Monte Carlo Methods

This chapter deepens the variance reduction subject, and focuses on the Monte Carlo methods for deterministic parabolic partial differential equations. This topic requires advanced notions in stochastic calculus, particularly the Girsanov theorem, which we state and discuss first. We strongly emphasize that universal techniques do not exist: most often, effective variance reduction methods depend on the numerical analyst’s knowledge and experience. We will see that it is rather easy to construct perfect variance reduction methods which are irrelevant from a numerical point of view; a contrario, the construction of an effective method often lies on the approximation of a perfect method, the approximation method needing to be adapted to each particular case. Interesting examples can be found in Duffie and Glynn (Ann. Appl. Probab. 5(4), 897–905, 1995).

Non-asymptotic Error Estimates for Monte Carlo Methods

The principles of Monte Carlo methods based on the Strong Law of Large Numbers (SLLN) are detailed. A number of examples are described, some of which correspond to concrete problems in important application fields. This is followed by the discussion and description of various algorithms of simulation, first for uniform random variables, then using these for general random variables. Eventually, the more advanced topic of martingale theory is introduced, and the SLLN is proved using a backward martingale technique and the Kolmogorov zero-one law.

Continuous-Space Markov Processes with Jumps

In order to effectively implement Monte Carlo methods, the random approximation errors must be controlled. For this purpose, theoretical results are provided for the estimation of the number of simulations necessary to obtain a desired accuracy with a prescribed confidence interval. Therefore absolute, i.e., non-asymptotic, versions of the Central Limit Theorem (CLT) are developed: Berry–Esseen’s and Bikelis’ theorems, as well as concentration inequalities obtained from logarithmic Sobolev inequalities. The difficult subject of variance reduction techniques for Monte Carlo methods arises naturally in this context, and is discussed at the end of this chapter.

Discrete-Space Markov Processes

From now on, Markov processes with continuous state space ((mathbb{R}^{d}) for some or one of its closed subsets) are considered. Their rigorous study requires advanced measure-theoretic tools, but we limit ourselves to developing the reader’s intuition, notably by pathwise constructions leading to simulations. We first emphasize the strong similarity between such Markov processes with constant trajectories between isolated jumps and discrete space ones. We then introduce Markov processes with sample paths following an ordinary differential equation between isolated jumps. In both cases, the Kolmogorov equations and Feynman–Kac formula are established. This is applied to kinetic equations coming from statistical Mechanics. These describe the time evolution of the instantaneous distribution of particles in phase space (position-velocity), when the particle velocity jumps at random instants in function of the particle position and velocity.

Stochastic Simulation and Monte Carlo Methods. Mathematical Foundations of Stochastic Simulation.

A rather detailed study of Markov processes with discrete state space is provided. It focuses on sample path techniques in a perspective inspired by simulation needs. The relationship of these processes with Poisson processes and with discrete-time Markov chains is shown. Rigorous constructions and results are provided for Markov process with uniformly bounded jump rates. To this end, elements of the theory of bounded operators are introduced, which explain the relation between generator and semigroup, and provide a useful framework for the forward and backward Kolmogorov equations and the Feynman–Kac formula.